| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s − 3·5-s + 9·6-s + 3·7-s + 10·8-s + 3·9-s − 9·10-s + 18·12-s + 3·13-s + 9·14-s − 9·15-s + 15·16-s + 3·17-s + 9·18-s − 18·20-s + 9·21-s − 15·23-s + 30·24-s + 6·25-s + 9·26-s + 27-s + 18·28-s − 3·29-s − 27·30-s − 9·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s + 1.13·7-s + 3.53·8-s + 9-s − 2.84·10-s + 5.19·12-s + 0.832·13-s + 2.40·14-s − 2.32·15-s + 15/4·16-s + 0.727·17-s + 2.12·18-s − 4.02·20-s + 1.96·21-s − 3.12·23-s + 6.12·24-s + 6/5·25-s + 1.76·26-s + 0.192·27-s + 3.40·28-s − 0.557·29-s − 4.92·30-s − 1.61·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24389000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24389000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.14516544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.14516544\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - p T + 2 p T^{2} - 10 T^{3} + 2 p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 4 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} - 24 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 40 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 42 T^{2} - 84 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 9 T^{2} - 56 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 15 T + 108 T^{2} + 24 p T^{3} + 108 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 9 T + 108 T^{2} + 556 T^{3} + 108 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 39 T^{2} + 232 T^{3} + 39 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 12 T + 99 T^{2} + 528 T^{3} + 99 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 144 T^{2} - 758 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 45 T^{2} + 192 T^{3} + 45 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 78 T^{2} - 468 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 15 T + 180 T^{2} + 1602 T^{3} + 180 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 15 T + 204 T^{2} - 1604 T^{3} + 204 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 18 T + 3 p T^{2} - 1676 T^{3} + 3 p^{2} T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 + 9 T + 234 T^{2} + 1312 T^{3} + 234 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 9 T + 102 T^{2} - 320 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 1920 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 243 T^{2} - 24 T^{3} + 243 p T^{4} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 282 T^{2} - 1748 T^{3} + 282 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.63558349084430931817762020080, −10.39576802160501566254891546059, −9.903128644917207584138040105758, −9.900102189037536620764853225375, −8.807995506409440880056456620375, −8.780491499792404110645782353211, −8.740768763493239935399568839108, −7.965254244668962494130985274267, −7.897885815155730349389946441766, −7.75234000453503620370712933227, −7.27335930412895876702869607926, −7.08820154662815520981181126867, −6.43990451466365518223286679974, −5.88306303986387319098906987374, −5.83393683358686715852853527128, −5.38584246802692058432397017240, −4.67807927963898269299364813226, −4.58550918474849529536313996909, −3.99536896728196627920557218645, −3.75702713742074221972192946275, −3.40347079805317736636773815143, −3.26772954396411075380503340706, −2.35928655279984995783914105728, −2.11697422565417834337721711706, −1.52241067737148877215906531093,
1.52241067737148877215906531093, 2.11697422565417834337721711706, 2.35928655279984995783914105728, 3.26772954396411075380503340706, 3.40347079805317736636773815143, 3.75702713742074221972192946275, 3.99536896728196627920557218645, 4.58550918474849529536313996909, 4.67807927963898269299364813226, 5.38584246802692058432397017240, 5.83393683358686715852853527128, 5.88306303986387319098906987374, 6.43990451466365518223286679974, 7.08820154662815520981181126867, 7.27335930412895876702869607926, 7.75234000453503620370712933227, 7.897885815155730349389946441766, 7.965254244668962494130985274267, 8.740768763493239935399568839108, 8.780491499792404110645782353211, 8.807995506409440880056456620375, 9.900102189037536620764853225375, 9.903128644917207584138040105758, 10.39576802160501566254891546059, 10.63558349084430931817762020080
